let X, Y be non empty TopSpace; :: thesis: for X0, X1 being non empty SubSpace of X

for g being Function of X0,Y st X1 is SubSpace of X0 holds

for A being Subset of X0

for x0 being Point of X0

for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

let X0, X1 be non empty SubSpace of X; :: thesis: for g being Function of X0,Y st X1 is SubSpace of X0 holds

for A being Subset of X0

for x0 being Point of X0

for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

let g be Function of X0,Y; :: thesis: ( X1 is SubSpace of X0 implies for A being Subset of X0

for x0 being Point of X0

for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )

assume A1: X1 is SubSpace of X0 ; :: thesis: for A being Subset of X0

for x0 being Point of X0

for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

let A be Subset of X0; :: thesis: for x0 being Point of X0

for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

let x0 be Point of X0; :: thesis: for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

let x1 be Point of X1; :: thesis: ( A is open & x0 in A & A c= the carrier of X1 & x0 = x1 implies ( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )

assume that

A2: ( A is open & x0 in A ) and

A3: A c= the carrier of X1 and

A4: x0 = x1 ; :: thesis: ( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

thus ( g is_continuous_at x0 implies g | X1 is_continuous_at x1 ) by A1, A4, Th74; :: thesis: ( g | X1 is_continuous_at x1 implies g is_continuous_at x0 )

thus ( g | X1 is_continuous_at x1 implies g is_continuous_at x0 ) :: thesis: verum

for g being Function of X0,Y st X1 is SubSpace of X0 holds

for A being Subset of X0

for x0 being Point of X0

for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

let X0, X1 be non empty SubSpace of X; :: thesis: for g being Function of X0,Y st X1 is SubSpace of X0 holds

for A being Subset of X0

for x0 being Point of X0

for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

let g be Function of X0,Y; :: thesis: ( X1 is SubSpace of X0 implies for A being Subset of X0

for x0 being Point of X0

for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )

assume A1: X1 is SubSpace of X0 ; :: thesis: for A being Subset of X0

for x0 being Point of X0

for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

let A be Subset of X0; :: thesis: for x0 being Point of X0

for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

let x0 be Point of X0; :: thesis: for x1 being Point of X1 st A is open & x0 in A & A c= the carrier of X1 & x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

let x1 be Point of X1; :: thesis: ( A is open & x0 in A & A c= the carrier of X1 & x0 = x1 implies ( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )

assume that

A2: ( A is open & x0 in A ) and

A3: A c= the carrier of X1 and

A4: x0 = x1 ; :: thesis: ( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

thus ( g is_continuous_at x0 implies g | X1 is_continuous_at x1 ) by A1, A4, Th74; :: thesis: ( g | X1 is_continuous_at x1 implies g is_continuous_at x0 )

thus ( g | X1 is_continuous_at x1 implies g is_continuous_at x0 ) :: thesis: verum

proof

assume A5:
g | X1 is_continuous_at x1
; :: thesis: g is_continuous_at x0

A is a_neighborhood of x0 by A2, CONNSP_2:3;

hence g is_continuous_at x0 by A1, A3, A4, A5, Th76; :: thesis: verum

end;A is a_neighborhood of x0 by A2, CONNSP_2:3;

hence g is_continuous_at x0 by A1, A3, A4, A5, Th76; :: thesis: verum